MECH3610 · Advanced Thermofluids
Radiation Heat Transfer
Radiation needs no medium and scales with the fourth power of absolute temperature, and this chapter extends the intro Stefan-Boltzmann law to full surface-to-surface exchange. It covers emissive power, the opaque-surface and Kirchhoff relations, view (configuration) factors with their reciprocity and summation rules, and blackbody and gray-surface exchange. Radiation combined with convection is a recurring exam theme, and view-factor algebra is examined on the mid-term and final.
What this chapter covers
- 01Emissive power E = epsilon*sigma*T^4 and the blackbody limit E_b = sigma*T^4 (temperatures in kelvin)
- 02Irradiation and the opaque-surface balance rho + alpha = 1; Kirchhoff's law epsilon = alpha
- 03View factor F_ij as the fraction of radiation leaving i that reaches j
- 04Reciprocity rule A_i*F_ij = A_j*F_ji
- 05Summation rule sum_j F_ij = 1 over an enclosure; self-view F_ii = 0 for plane/convex surfaces
- 06Blackbody exchange Q_ij = A_i*F_ij*sigma*(T_i^4 - T_j^4)
- 07Net radiation from a surface Q_i = sum_j A_i*F_ij*sigma*(T_i^4 - T_j^4)
- 08Gray-surface radiosity network: surface resistance (1 - epsilon)/(epsilon*A) and space resistance 1/(A_i*F_ij)
View-factor reciprocity and blackbody exchange between two surfaces
- +1Reciprocity gives A1*F12 = A2*F21, so F21 = A1*F12/A2 = 0.1*0.25/0.4 = 0.0625.
- +1For two black surfaces the net exchange is Q12 = A1*F12*sigma*(T1^4 - T2^4), with sigma = 5.67e-8 W/m2-K4 and absolute temperatures.
- +1T1^4 = 800^4 = 4.096e11 and T2^4 = 400^4 = 2.56e10, so the difference is 3.84e11. Then Q12 = 0.1*0.25*5.67e-8*3.84e11 = 0.025*5.67e-8*3.84e11 = 544 W.
Key terms
- Emissive power (E)
- The radiation a surface emits per unit area, E = epsilon*sigma*T^4 [W/m2] (kelvin). The blackbody limit is E_b = sigma*T^4 with emissivity epsilon = 1.
- Kirchhoff's law
- At thermal equilibrium a surface's spectral emissivity equals its spectral absorptivity (epsilon = alpha). For a gray surface this holds over all wavelengths, simplifying radiation exchange.
- View factor (F_ij)
- The fraction of radiation leaving surface i that arrives directly at surface j, set purely by geometry. It obeys reciprocity and enclosure-summation rules.
- Reciprocity rule
- A_i*F_ij = A_j*F_ji: a geometric identity that lets one view factor be found from its partner and the two areas.
- Summation rule
- For an enclosure, sum_j F_ij = 1 — all radiation leaving surface i must land on some surface of the enclosure (including itself if concave; F_ii = 0 for plane or convex surfaces).
- Radiosity network
- A resistance model for gray-surface exchange using a surface resistance (1 - epsilon)/(epsilon*A) at each surface and a space resistance 1/(A_i*F_ij) between surfaces, solved like a thermal circuit.
Radiation Heat Transfer FAQ
Do radiation calculations need Kelvin?
Always. Emissive power and every exchange relation contain T^4, so any temperature inside a radiation term must be absolute (kelvin). Converting only after the fourth power, or leaving Celsius inside, produces wildly wrong answers — this is the single most common radiation mistake.
How do I find view factors I am not given?
Use the two algebra rules. Reciprocity A_i*F_ij = A_j*F_ji converts a known factor into its partner, and the enclosure summation sum_j F_ij = 1 closes the set. For plane or convex surfaces F_ii = 0 (a surface cannot see itself); concave surfaces have F_ii > 0. Standard charts supply the primary factors for common geometries.
What is the difference between blackbody and gray-surface exchange?
Black surfaces absorb and emit perfectly (epsilon = 1), so exchange is simply Q_ij = A_i*F_ij*sigma*(T_i^4 - T_j^4). Real gray surfaces reflect some radiation, so you add surface resistances (1 - epsilon)/(epsilon*A) and solve a radiosity network — the same resistance-circuit idea used for conduction.
How is radiation examined in MECH3610?
As view-factor algebra and exchange calculations, and — importantly — as combined convection-plus-radiation surface balances (the steam-pipe and small-object-in-surroundings problems from earlier chapters). The equation sheet carries the view-factor and exchange relations for the open-book exam. Confirm the coverage on Moodle.
Exam move
Anchor everything on the fourth-power law and kelvin. Practise the view-factor algebra — reciprocity and enclosure summation — until you can complete a small enclosure's factors quickly, and keep the blackbody exchange relation and the gray-surface resistances side by side so you can decide which the problem needs. Rehearse the combined convection-plus-radiation surface balance, since that is where radiation most often appears in this course, and remember the linearised h_r trick to fold radiation into a Newton-cooling parallel resistor. Confirm the exam format and permitted charts on Moodle.
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